80.11.15 problem 14 (b)
Internal
problem
ID
[21423]
Book
:
A
Textbook
on
Ordinary
Differential
Equations
by
Shair
Ahmad
and
Antonio
Ambrosetti.
Second
edition.
ISBN
978-3-319-16407-6.
Springer
2015
Section
:
Chapter
12.
Stability
theory.
Excercise
12.6
at
page
270
Problem
number
:
14
(b)
Date
solved
:
Thursday, October 02, 2025 at 07:31:08 PM
CAS
classification
:
system_of_ODEs
\begin{align*} x_{1}^{\prime }\left (t \right )&=x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=x_{3} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=-a x_{3} \left (t \right )-b x_{2} \left (t \right )-c x_{1} \left (t \right ) \end{align*}
✓ Maple. Time used: 0.183 (sec). Leaf size: 3863
ode:=[diff(x__1(t),t) = x__2(t), diff(x__2(t),t) = x__3(t), diff(x__3(t),t) = -a*x__3(t)-b*x__2(t)-c*x__1(t)];
dsolve(ode);
\begin{align*}
\text {Expression too large to display} \\
x_{2} \left (t \right ) &= c_1 \,{\mathrm e}^{-\frac {\left (i \left (36 b a -108 c -8 a^{3}+12 \sqrt {12 c \,a^{3}-3 b^{2} a^{2}-54 b a c +12 b^{3}+81 c^{2}}\right )^{{2}/{3}} \sqrt {3}-4 i \sqrt {3}\, a^{2}+12 i \sqrt {3}\, b +\left (36 b a -108 c -8 a^{3}+12 \sqrt {12 c \,a^{3}-3 b^{2} a^{2}-54 b a c +12 b^{3}+81 c^{2}}\right )^{{2}/{3}}+4 a \left (36 b a -108 c -8 a^{3}+12 \sqrt {12 c \,a^{3}-3 b^{2} a^{2}-54 b a c +12 b^{3}+81 c^{2}}\right )^{{1}/{3}}+4 a^{2}-12 b \right ) t}{12 \left (36 b a -108 c -8 a^{3}+12 \sqrt {12 c \,a^{3}-3 b^{2} a^{2}-54 b a c +12 b^{3}+81 c^{2}}\right )^{{1}/{3}}}}+c_2 \,{\mathrm e}^{\frac {\left (i \left (36 b a -108 c -8 a^{3}+12 \sqrt {12 c \,a^{3}-3 b^{2} a^{2}-54 b a c +12 b^{3}+81 c^{2}}\right )^{{2}/{3}} \sqrt {3}-4 i \sqrt {3}\, a^{2}+12 i \sqrt {3}\, b -\left (36 b a -108 c -8 a^{3}+12 \sqrt {12 c \,a^{3}-3 b^{2} a^{2}-54 b a c +12 b^{3}+81 c^{2}}\right )^{{2}/{3}}-4 a \left (36 b a -108 c -8 a^{3}+12 \sqrt {12 c \,a^{3}-3 b^{2} a^{2}-54 b a c +12 b^{3}+81 c^{2}}\right )^{{1}/{3}}-4 a^{2}+12 b \right ) t}{12 \left (36 b a -108 c -8 a^{3}+12 \sqrt {12 c \,a^{3}-3 b^{2} a^{2}-54 b a c +12 b^{3}+81 c^{2}}\right )^{{1}/{3}}}}+c_3 \,{\mathrm e}^{\frac {\left (\left (36 b a -108 c -8 a^{3}+12 \sqrt {12 c \,a^{3}-3 b^{2} a^{2}-54 b a c +12 b^{3}+81 c^{2}}\right )^{{2}/{3}}-2 a \left (36 b a -108 c -8 a^{3}+12 \sqrt {12 c \,a^{3}-3 b^{2} a^{2}-54 b a c +12 b^{3}+81 c^{2}}\right )^{{1}/{3}}+4 a^{2}-12 b \right ) t}{6 \left (36 b a -108 c -8 a^{3}+12 \sqrt {12 c \,a^{3}-3 b^{2} a^{2}-54 b a c +12 b^{3}+81 c^{2}}\right )^{{1}/{3}}}} \\
\text {Expression too large to display} \\
\end{align*}
✓ Mathematica. Time used: 0.007 (sec). Leaf size: 488
ode={D[x1[t],t]==x2[t],D[x2[t],t]==x3[t],D[x3[t],t]==-a*x3[t]-b*x2[t]-c*x1[t]};
ic={};
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
\begin{align*} \text {x1}(t)&\to c_3 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2 a+\text {$\#$1} b+c\&,\frac {e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1} a+b}\&\right ]+c_2 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2 a+\text {$\#$1} b+c\&,\frac {a e^{\text {$\#$1} t}+\text {$\#$1} e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1} a+b}\&\right ]+c_1 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2 a+\text {$\#$1} b+c\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}+\text {$\#$1} a e^{\text {$\#$1} t}+b e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1} a+b}\&\right ]\\ \text {x2}(t)&\to -c c_1 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2 a+\text {$\#$1} b+c\&,\frac {e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1} a+b}\&\right ]+c_3 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2 a+\text {$\#$1} b+c\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1} a+b}\&\right ]+c_2 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2 a+\text {$\#$1} b+c\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}+\text {$\#$1} a e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1} a+b}\&\right ]\\ \text {x3}(t)&\to -c c_1 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2 a+\text {$\#$1} b+c\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1} a+b}\&\right ]+c_3 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2 a+\text {$\#$1} b+c\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1} a+b}\&\right ]-c_2 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2 a+\text {$\#$1} b+c\&,\frac {\text {$\#$1} b e^{\text {$\#$1} t}+c e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1} a+b}\&\right ] \end{align*}
✗ Sympy
from sympy import *
t = symbols("t")
a = symbols("a")
b = symbols("b")
c = symbols("c")
x1 = Function("x1")
x2 = Function("x2")
x3 = Function("x3")
ode=[Eq(-x2(t) + Derivative(x1(t), t),0),Eq(-x3(t) + Derivative(x2(t), t),0),Eq(a*x3(t) + b*x2(t) + c*x1(t) + Derivative(x3(t), t),0)]
ics = {}
dsolve(ode,func=[x1(t),x2(t),x3(t)],ics=ics)
Timed Out